When a line in the coordinate plane is dilated with respect to the origin, each point on the line is transformed based on the scale factor.
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Effects of Dilation with a Scale Factor of 3:
- Each point \( (x, y) \) on line AB will move to a new point \( (3x, 3y) \) on line A'B'.
- The direction of the line remains unchanged because dilation with respect to the origin preserves angles and straightness.
- The distance of the line from the origin increases, as every point is moved away from the origin by a factor of 3.
- The new line A'B' will still be parallel to the original line AB, but it will be three times farther from the origin. Therefore, the steepness or slope of the line remains the same, but the line is scaled outward from the origin.
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Effects of Dilation with a Scale Factor of 13:
- Similarly, if the scale factor is increased to 13, each point \( (x, y) \) on line AB will move to \( (13x, 13y) \) on line A'B'.
- Just like before, the direction and slope of the line remain unchanged.
- The distance of the line from the origin increases even further, with each point being moved 13 times farther away from the origin compared to its original position.
- The new line A'B' will still be parallel to the original line AB, but at this scale factor, it will be positioned even further from the origin than with the scale factor of 3.
Summary: Both dilations maintain the line's slope and parallelism to the original line while increasing the distance from the origin. The larger the scale factor, the further the line moves away from the origin. Thus, the key difference between the two dilations is the degree of scaling: a factor of 3 results in points being tripled in distance from the origin, while a factor of 13 results in points being 13 times further from the origin.
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